In this paper, the smallest confidence region is obtained for the location and scale parameters of the two-parameter exponential distribution. For this purpose, we use constrained optimization problems. We first provide some suitable pivotal quantities to obtain a balanced confidence region. We then obtain the smallest confidence region by minimizing the area of the confidence region using the Lagrangian method. Two numerical examples are presented to illustrate the proposed methods. Finally, some applications of proposed joint confidence regions in hypothesis testing and the construction of confidence bands are discussed.

In this paper, based on an appropriate pivotal quantity, two methods are introduced to determine confidence region for the mean and standard deviation in a two parameter uniform distribution, in which the application of numerical methods is not mandatory. In the first method, the smallest region is obtained by minimizing the confidence region's area, and in the second method, a simultaneous Bonferroni confidence interval is introduced by using the smallest confidence intervals. By the comparison of area and coverage probability of the introduced methods, as well as, comparison of the width of strip including the standard deviation in both methods, it has been shown that the first method has a better efficiency. Finally, an approximation for the quantile of F
distribution used in calculating the confidence regions in a special case is presented.